AI 中文总结
针对噪声下二维映射，该研究提出有限时间混沌诊断框架，证明最大FTLE的高斯收敛性，开发吸引子分离算法并在Domenicali映射上验证，确定吸引子合并的临界噪声水平。
AI 中文摘要
从气候模型到电网，现实系统常因传感器噪声、漂移或环境变异性发生波动，而标准混沌诊断假设参数固定且具有渐近视界。我们针对独立同分布参数噪声下的二维映射引入有限时间框架：首先，证明经中心化和缩放后的最大有限时间李雅普诺夫指数收敛于高斯分布，其均值与方差明确依赖于映射的雅可比统计量；其次，开发一种吸引子分离算法，利用有限时间李雅普诺夫指数（FTLE）直方图和基于几何的分类器，在噪声下将相空间划分为混沌与周期区域；第三，在带噪声的Domenicali映射上对理论进行数值验证，证明FTLE服从高斯分布、Kaplan-Yorke维数发生可预测偏移，且存在明确的吸引域逃逸噪声阈值；最后，采用三种互补数值方法，对吸引子合并发生的临界噪声水平σ_c进行上界估计与精准定位。
英文摘要
Real-world systems, from climate models to power grids, often fluctuate due to sensor noise, drift, or environmental variability, yet standard chaos diagnostics assume fixed parameters and asymptotic horizons. We introduce a finite-time framework for two-dimensional maps under independent, identically distributed parameter noise. First, we prove that the maximal finite-time Lyapunov exponent converges, after centering and scaling, to a Gaussian law whose mean and variance depend explicitly on the map's Jacobian statistics. Second, we develop an attractor separation algorithm that uses FTLE histograms and a geometry-based classifier to partition phase space into chaotic and periodic regions under noise. Third, we validate our theory numerically on the noisy Domenicali map, demonstrating Gaussian FTLE distributions, predictable shifts in Kaplan-Yorke dimension, and a sharp noise threshold for basin escape. Finally, we estimate the critical noise level $σ_c$ at which attractor coalescence occurs, using three complementary numerical methods to bound it above and to pinpoint an estimate.
Comments38 pages, 13 figures
